Questions tagged [set-theory]
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
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In the construction of $K^c$ for sequences of measures, why do we only add measures of cofinality $\omega_1$?
$\require{cancel}$
I have a question about the definition of $K^c$ for sequences of measures in Mitchell's "The Covering Lemma" chapter in the Handbook. I will give the definition he gives ...
- 185
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Natural functions outside $\sf PA$?
Can theories stronger than $\sf PA$ manage to define functions from the naturals to the naturals, that $\sf PA$ cannot? If so, what are examples of those functions?
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At what ordinal $\chi$ does $\mathrm{L}_\chi$ contain a surjection from $\omega$ to $\mathrm{L}_{\beta_0}$?
Let $\mathrm{ZF^-}$ be $\mathrm{ZF}$ minus power set, and let $\beta_0$ be the ordinal for ramified analysis so that $\mathrm{L}_{\beta_0}$ is the least $\mathrm{L}$-model of $\mathrm{ZF}^-$. Clearly, ...
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Non-computable numbers in constructive mathematics
Edited in order to take into account feedback from comments:
If we have an uncountable formal language L with well-defined semantics, can every set in the set-theoretic universe V be explicitly ...
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Can PA define functions related to higher theories?
Working in $\sf ZFC$ we can define a function $F$ from naturals to naturals such that $F(0) = \, ^\circ\ulcorner r({\sf Z}) \urcorner$, where $r({\sf Z})$ is the Rosser sentence of Zermelo set theory $...
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Large almost disjoint family on $\mathbb{N}$ with property $\mathbf{B}$
Let
$\newcommand{\oo}{[\omega]^\omega}\oo$ denote the collection of all infinite subsets of the set of nonnegative integers $\omega$. We say that $\newcommand{\ss}{{\cal S}}\S\subseteq \oo$ ...
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64
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Axiom of choice for function domain reconstruction
I know the following definition of the axiom of choice:
For each family $S$ of nonempty disjoint sets, there is a set $V$ (the
so-called selector) that contains exactly one element from each of the
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Can a finitely axiomatized theory be equivalently axiomatized in a proper infinite manner?
This comes in continuation to this prior question about infinite axiomatizability.
let $T$ be a finitely axiomatizable first order set (no class ontology) theory that can interpret MacLane set theory. ...
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802
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Is there a minimal (least?) countably saturated real-closed field?
I heard from a reputable mathematician that ZFC proves that there is a minimal countably saturated real-closed field. I have several questions about this.
Is there a soft model-theoretic construction ...
- 230k
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85
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Consistency of the Rothberger property [closed]
Are there any models with this property present? With large cardinals? I am having trouble finding it though there's papers showing that its equivalent to other notions.
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A function $f$ such that $j_U(f)(\kappa)=[\operatorname{id}]_U$ for all ultrapower embeddings $j_U$ with critical point $\kappa$
Let $\kappa$ be a measurable cardinal. Is there a function $f\colon\kappa\to V$ such that whenever $j_U\colon V\to\operatorname{Ult}(V,U)$ is an ultrapower embedding with critical point $\kappa$, we ...
- 1,479
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102
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Coherent sequence of ultrafilters in iterated forcing extensions
Remember that if $\kappa$ is strongly compact, then any ${<}\kappa$-complete filter extends to a ${<}\kappa$-complete ultrafilter.
Let $\Bbb P_\delta=\langle\Bbb P_\alpha,\dot{\Bbb Q}_\alpha\mid ...
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Mostowski's absoluteness theorem and proving that theories extending $0^\#$ have incomparable minimal transitive models
This question says that the theory ZFC + $0^\#$ has incomparable minimal transitive models. It proves this as follows (my emphasis):
[F]or every c.e. $T⊢\text{ZFC\P}+0^\#$ having a model $M$ with $On^...
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Can this formalism prove the consistency of ZFC?
Working in bi-sorted first order logic with equality and membership. First sort variables written in lower case range over sets. Second sort variables written in upper case range over classes. Lets ...
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Whence compactness of automorphism quantifiers?
The following question arose while trying to read Shelah's papers Models with second-order properties I-V. For simplicity, I'm assuming a much stronger theory here than Shelah: throughout, we work in $...
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